When does a linear map belong to at least one orthogonal or symplectic group?

Abstract

Given an endomorphism u of a finite-dimensional vector space over an arbitrary field K, we give necessary and sufficient conditions for the existence of a regular quadratic form (respectively, a symplectic form) for which u is orthogonal (respectively, symplectic). Since a solution to this problem is already known in the case char(K)≠2, our main contribution lies in the case char(K)=2. When char(K)=2, we also give necessary and sufficient conditions for the existence of a regular symmetric bilinear form for which u is orthogonal. When K is finite with characteristic 2, we give necessary and sufficient conditions for the existence of an hyperbolic quadratic form (respectively, a regular non-hyperbolic quadratic form, respectively, a regular nonalternate symmetric bilinear form) for which u is orthogonal.